Object rotates when Copy Location Constraint enabled
So let me make sure I understand what's happening: through frame 100 (ish, can't see really well) you're using a child-of constraint, then at frame 101 you set the influence to 0 of the child-of, and set the influence of the copy location constraint to 1.0?
It's not the copy location constraint that is rotating the object.  It's the fact that the object is no longer under the influence of the child-of constraint.  At frame 100, that child of constraint is creating rotation on the object, inherited from the target of the child of.  At frame 101, that constraint becomes inactive, and it loses that rotation all at once.
How to fix? If you haven't already, start by keyframing rotation for the object on frame 100.  Then, go to frame 101.  Set the influence of the child-of constraint to 1.0.  In the 3D window, apply visual transform (from ctrl-a menu for me) and keyframe rotation.  The object will jump momentarily; don't worry about it.  Then set the influence of the child-of constraint back to 0.0.  The object will have the same rotation it would have as if it were a child-of, but without needing the child-of to get it.